Mathematical Sciences: Function Theory in Several Complex Variables

黄黄将研究复空间中实超曲面的映射问题，并尝试证明Cauchy-Riemann映射的各种正则性结果。他将研究复空间中实子流形的光滑和解析结构及其全纯局部壳的结构。他还将探讨复n空间中有界域的Kobayashi极值映射及其在全纯自映射研究中的应用。本世纪初出现了几个复变量，作为一个复变量函数研究的自然产物。很明显，这个理论与它的前身有很大的不同。底层的几何结构更难掌握，而函数理论与一阶偏微分算子的关系要密切得多。因此，它成长为一门混合学科，结合了微分几何和微分方程的深层特征。许多基本结构是在过去三十年中确定的。目前的研究仍然集中在理解这些基本的数学形式。

DMS-9500881 Huang Huang will study the mapping problems for real hypersurfaces in complex spaces and try to prove various regularity results for Cauchy-Riemann mappings. He will investigate the smooth and analytic structure of a real submanifold in the complex space and the structure of its local hull of holomorphy. Also he will explore the Kobayashi extremal mappings of bounded domains in complex n-space and their applications to the study of holomorphic self-mappings. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms.